Sufficient dimension reduction for average causal effect estimation

A large number of covariates can have a negative impact on the quality of causal effect estimation since confounding adjustment becomes unreliable when the number of covariates is large relative to the number of samples. Propensity score is a common way to deal with a large covariate set, but the accuracy of propensity score estimation (normally done by logistic regression) is also challenged by the large number of covariates. In this paper, we prove that a large covariate set can be reduced to a lower dimensional representation which captures the complete information for adjustment in causal effect estimation. The theoretical result enables effective data-driven algorithms for causal effect estimation. Supported by the result, we develop an algorithm that employs a supervised kernel dimension reduction method to learn a lower dimensional representation from the original covariate space, and then utilises nearest neighbour matching in the reduced covariate space to impute the counterfactual outcomes to avoid the large sized covariate set problem. The proposed algorithm is evaluated on two semisynthetic and three real-world datasets and the results show the effectiveness of the proposed algorithm.